# How do you differentiate y= (5x)/((tanx)(cotx))?

Apr 13, 2015

5

tan x * cot x is simply 1, because cot x equals $\frac{1}{\tan} x$. Thus it is y= 5x. Hence $\frac{\mathrm{dy}}{\mathrm{dx}}$ =5

Apr 13, 2015

If you don't think first (before you start) you'll use the quotient rule (with the product rule to differentiate the denominator).

Take a few seconds to think, and to ask:
Can I rewrite this before I differentiate to make my life easier?

$\tan x = \frac{1}{\cot} x$ so $\tan x \cot x = 1$

That means that $y = 5 \frac{x}{1} = 5 x$

So, $y ' = 5$

Apr 13, 2015

$y ' = 5$

y=(5x)/((tanx)(cotx)

As

$\tan x = \sin \frac{x}{\cos} x$

And

$\cot x = \cos \frac{x}{\sin} x$

So,

$y = \frac{5 x}{\left(\sin \frac{x}{\cos} x\right) \left(\cos \frac{x}{\sin} x\right)}$

$y = \frac{5 x}{\left(\cancel{\sin} \frac{x}{\cancel{\cos}} x\right) \left(\cancel{\cos} \frac{x}{\cancel{\sin}} x\right)}$

$y = \frac{5 x}{1}$

$y = \left(5 x\right)$

Differentiating both sides with respect to 'x'

$y ' = 5$